Ambiguity in matter sector for modified gravity involving $$\delta ^2 \mathcal {L}_{m}/\delta g^{\mu \nu }\delta g^{\alpha \beta }$$ and its implications to astrophysics and cosmology

Abstract

Matter density (

) and radial pressure ( p ) are often used as the matter Lagrangian density (

$$\mathcal {L}_{m}$$ L m

) because both are thermodynamically consistent and produce the same Einstein field equation (EFE) in general relativity (GR). New gravity models with explicit links between matter and geometry instead involve second-order derivatives of

$$\mathcal {L}_{m}$$ L m

relative to the metric tensor. So, picking either p or

$$-\rho $$ - ρ

for

$$\mathcal {L}_{m}$$ L m

gives different effective EFEs. This confusion appears because one usually treats the four-velocity (

$$u_\mu $$ u μ

) and the metric tensor (

$$g_{\mu \nu }$$ g μ ν

) as independent. Here, we revisit the basics and offer a consistent framework by relaxing that assumption, thereby making the modified gravity theory independent of the choice of

$$\mathcal {L}_{m}$$ L m

. Finally, we test this approach on neutron and quark stars (ultraviolet region) and on cosmological situations with matter-dominated or dust (

$$p=0$$ p = 0

), radiation-dominated (

$$p=\rho /3$$ p = ρ / 3

), and vacuum energy (

$$p=-\rho $$ p = - ρ

) equations of state (infrared region), showing how it clarifies the ambiguity in picking

$$\mathcal {L}_{m}$$ L m

for gravity models.